Partly melted DNA conformations obtained with a probability peak finding method

Peaks in the probabilities of loops or bubbles, helical segments, and unzipping ends in melting DNA are found in this article using a peak finding method that maps the hierarchical structure of certain energy landscapes. The peaks indicate the alternative conformations that coexist in equilibrium and the range of their fluctuations. This yields a representation of the conformational ensemble at a given temperature, which is illustrated in a single diagram called a stitch profile. This article describes the methodology and discusses stitch profiles vs the ordinary probability profiles using the phage lambda genome as an example.

Figure 1

A stitch profile is composed of four types of stitches: (a) loops, (b) right tails, (c) left tails, and (d) helices. Loop and tail stitches are drawn on the upper side and they span regions of opened base pairs (0’s). Helix stitches are drawn on the lower side and they span regions of closed base pairs (1’s). The horizontal bars indicate fluctuational ranges of the 0-1 boundaries.

Figure 2

(Color online) $E_1\left(x\right)$ is plotted for a 70 bp sequence to illustrate the pedigree ordering of lakes in an energy landscape. Lakes corresponding to each local maximum are shown as horizontal dashed (blue) lines. Arrows indicate the binary tree starting from the root ${\rho }_1=4$. Paternal lines are connected series of fathers shown as solid (red) arrows. The node $a=16$, for example, has the lake $L\left(a\right)=$ [15, 28], bottom $\beta a=20$, depth $D\left(a\right)\simeq 0.8$, father $\pi a=18$, mother $\mu a=15$, and successor $\sigma a=28$, as shown.

Figure 3

The MAXDEEP algorithm that finds stitches with depths below $D_{\max }$. For a given input node $i$, it returns a list of those nodes in $\Delta \left(i\right)$ that fulfill the criterion in Eq. (15). $c$ is the tree climber, → means push (i.e., “put on a list”), and $I\left(\mu c\right)$ is the output of a recursive call to the algorithm itself with $\mu c$ as the input node. The diagram can be read in conjunction with Fig. 2.

Figure 4

Lakes in 1D found by the MAXDEEP algorithm. The two plots show the same region of an energy landscape. The first plot shows the lakes found by the algorithm using $D_{\max }=3$ and the second plot shows the lakes with $D_{\max }=6$.

Figure 5

Eight quantities characterize a loop stitch.

Figure 6

A stitch profile (in the box) represents a number of alternative conformations. In this example, there are seven possible conformations listed schematically below the box. The parameters are $D_{\max }=3$, $p_{\mathrm{c}}=0.02$, and $T=81.9°\mathrm{C}$ and the sequence window is $35.5–37\mathrm{kbp}$.

Figure 7

(Color online) Comparison of a stitch profile and a probability profile, both calculated at $T\simeq 90°\mathrm{C}$ where the helicity is $\Theta =0.1$. The curve in the middle (in red) is the probability profile $1-p_{\mathrm{bp}}\left(x\right)$ and it varies between 0 and 1 (vertical axis not shown). Each stitch is labeled with its peak volume $p_{\mathrm{v}}$ in percent. The parameters are $D_{\max }=3$ and $p_{\mathrm{c}}=0.02$ and the sequence window is $11–17.4\mathrm{kbp}$.

Figure 8

(Color online) Extra stitches are included when the cutoff $p_{\mathrm{c}}$ is lowered. The first stitch profile $(p_{\mathrm{c}}=0.02)$ contains stitches with peak volumes $p_{\mathrm{v}}⩾0.02$. The second stitch profile $(p_{\mathrm{c}}=0.001)$ contains the same stitches as the first one (in blue), plus an extra set of stitches with peak volumes $0.001⩽p_{\mathrm{v}}<0.02$ (in red). The other parameters are $D_{\max }=3$ and $T\simeq 80.6°\mathrm{C}$ and the sequence window is $24–27.7\mathrm{kbp}$.

Figure 9

(Color online) The probability profile $1-p_{\mathrm{bp}}\left(x\right)$ of Fig. 7 is plotted here (in red) with its upper and lower bounds (in blue), $1-p_{\mathrm{low}}\left(x\right)$ and $1-p_{\mathrm{up}}\left(x\right)$, obtained from the stitch profile.